Finding $X$ from $X \left(\begin{smallmatrix} a \\ b \\c \end{smallmatrix}\right) = \left(\begin{smallmatrix} b \\ c \\a \end{smallmatrix}\right)$

Question

I am trying to solve the following matrix equation.

$$X \begin{pmatrix} a \\ b \\c \end{pmatrix} = \begin{pmatrix} b \\ c \\a \end{pmatrix}$$

My analysis:

• $X$ must be 3 by 3 matrix.
• $X=\begin{pmatrix} b \\ c \\a \end{pmatrix} \begin{pmatrix} a \\ b \\c \end{pmatrix}^{-1}$

Question

How can I calculate $ \begin{pmatrix} a \\ b \\c \end{pmatrix}^{-1} $?

Answer 1

You can't; that matrix has no inverse.

On the other hand, you know that$$X.\begin{bmatrix}1\\0\\0\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix}$$and therefore the first column of $X$ will be $\left[\begin{smallmatrix}0\\0\\1\end{smallmatrix}\right]$. You can compute the other columns by the same method. You will get that$$X=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}.$$

Answer 2

I don't know how to put it, but it's basically a permutation matrix: $$X=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}$$

Answer 3

$X$ belongs to a special family of matrices called permutation matrices, which swap elements of the input. For more information, see here: https://en.wikipedia.org/wiki/Permutation_matrix